Completing Einstein's Agenda

Cloud Cosmology

I am today sharing with you on my 65^th birthday, the unpublished second part of my paper published in AIP's Physics Essays June 2010. I will if invited, consider submitting it to a professional journal although I consider this venue sufficient for active review and inviting comment and interesting questions that can enhance the presentation. 

 In fact I invite suggestions and input.  The few equations did not post so you may wish to have me email you a clean copy.  Ask by emailing me cat arclein@yahoo.com.  This work leads to a slew of additional conjectures and are worth while cataloging and I do not mind assigning attribution here even when they are already obvious to myself and even thought of.  

As a teenager, I understood that the important theoretical question that needed to be answered was the melding of a foundational mathematical model of particle physics into Einstein's foundational mathematical model of General Relativity. This was Einstein's core agenda. Of course, I entered the field at about the same time that this effort was been abandoned by theorists and physicists generally in favor of the empirically rewarding quantum theory which is a Ptolemaic approach rather than foundational mathematics.

Regardless I received valuable guidance and insight and have spent the succeeding years defining and investigating many problems of interest. This blog is part of that ongoing process. As well I received insight that the results published here could best wait until I turned sixty five. There are a couple of reasons that this might be true. To start with, the present approach has utterly bogged down into a theoretical quagmire that is only now been admitted. It is time for change. The second and far more important reason is that our computer power was way to weak to properly run the simulations that this work begs. There really was no rush.

Claim 1 The advent of generalized cyclic functions into the field of mathematics allows us to expand our mathematica hugely into higher order solutions. An additional tweak of symbolic logic also holds out the plausible proposition that a great deal of our core mathematica can be mapped and confirmed by computer simulation. In short it is possible to tidy up the foundations of our mathematica and get a convincing computer assist. This leads to a conjecture.

Conjecture: Godel's Theorem can be voided.

I will be interjecting additional comments and interpretations throughout this essay that are not part of the original text and they will be normally in the form of conjectures. This hopefully will make this essay easier to read. However, the work on the particle pair in the earliest going is excessively rigorous as it has to be. You can expect to get a headache as you master it because no word is unexamined, not can be slid over easily. Go carefully and take your time.

Claim 2 The first creation of the particle pair fp described herein is sufficient to create the cosmos and all of time and space or the space time manifold and thereby unites Particle Physics and General Relativity.

Conjecture: A review of the empirical data is sufficient to suggest that the foundational fp described in this paper is the neutral neutrino and is a sufficient explanation for the content of the cosmos and of Dark Matter in particular. It may also be necessary and unique and in fact subject to mathematical proof. The lack of perfect packing for tetrahedrons strongly suggests just that.

Conjecture: The creation of a neutral neutrino is offset by the simultaneous creation of an anti neutral neutrino formed as a twist and counter twist. From that point they break apart and cannot cancel each other out unless they are able to realign. This is extremely unlikely unless tight packing is achieved. This framework produces our cosmological content without a net input of effort beyond initiation.

Claim 3 The theory generates a superior cosmology which we will now name as Cloud Cosmology. The act of creation produces an expanding sphere of partially bounded curvature geometries that decay readily into particle pairs fps when conditions are sufficient. This produces a cosmological cloud of neutral neutrinos and a resultant background radiation reflecting the decay process. The neutral neutrino does not produce gravity but that changes as assembly takes place and decay produces much larger particles and more energetic radiation. Assembly is the result of random perturbation and correct alignment leading to attractive combinations.

Conjecture: Self assembly produces a background of proto particles that will ultimately decay into all the particles that we recognize and related radiation. These particles are recognized as cosmic radiation and hydrogen. Hydrogen is formed as two neutrons self assemble and then decay to produce our hydrogen atom.

After all that takes place general gravitational effects will produce the rest of our present cosmos as understood.

As I state in the paper, my only assumption is that we exist. The act of creation itself is an act of imagination that includes a simple twist and plausibly its counter twist to initiate a space time manifold and the first particle pair(s) fp. Everything else follows at the internal speed of light. What is truly unimaginable is that we can imagine this.

Claim 4 Neutral neutrinos and anti neutral neutrinos will self assemble to form up as proto assemblages that will decay into geometric solids recognized in part through our standard theory. The electron is plausibly a solid holding about 600 such neutrinos. There is ample room for variation here and neutrino sharing. Simulation will need to investigate a range of options to determine stability. We can extend this same idea to assemble electron anti electron pairs or something similar to form the neutron which is central to our perspective of matter. We do not assume that these geometric solids are perfect.

Conjecture: A neutron contains over 720000 neutrinos of both types. Our metric allows us to map the net induced curvature inside and outside this assemblage for any point we choose. This will allow an appreciation of the power of the metric and the computation challenge involved to map the induced fields. It is now possible.

To put the computational problem into perspective, each point that is sampled for a neutron will require over 7200000 convergence calculations that need to be consistently stable in an environment that is deteriorating as the inverse of distance from the center. Sooner or later it be comes unstable and crash.

It is also clear that if we can achieve that, it will be possible to generate all possible smaller assemblages and their induced curvature.

Paper Begins

Before we begin our discussion of the implications of the generalized cyclic function for physics, a few remarks are in order about the mathematics and what we are attempting to achieve in the balance of the paper and to provide a bit of a road map to the reader. At least you will know what I think is important and what is mere speculation at this time.

The mathematics is a direct expansion of the idea of complex number through the application of a non reducibility rule for the nth root of -1 for all n greater than 2. An immediate and direct result is the generation of the generalized cyclic function of order greater than two. That encouraged the further development of the Pythagorean identities for orders three and four. It is also clear that higher orders of the Pythagorean identities can be developed to order n with the application of significant time and effort. The calculations are simple as demonstrated in the worked cases of the third and forth order.

The existence of Pythagorean identities of a higher order than two is new to mathematics and previously unimagined. I make the conjecture that it represents a powerful new tool allowing the orderly development of exact solutions in higher ordered differential equations. An obvious problem of some fame is the three, four and n body problem in mathematical physics. The new Pythagorean identities provides us with tools needed to attempt resolution.

I do not expect an easy or even a complete solution to arise, but I do know that by first reframing the historic work on the two body problem in terms of the second order Pythagorean, that it is plausible to extend the resultant ideas into the three body problem with some confidence. Much of this has been done and it will be valuable to establish best notation and general framework before this program is launched.

The generalized cyclic functions themselves are congruent in the sense that the curves maintain the same shape inside an envelope formed by the exponential function over +/- Y while the cycle length (or wavelength) for n > 2 declines as the inverse of the number of cycles. This is not yet formally derived but apparent as the individual curves are mapped from the spiral function. Natural congruence of the curves for all n implies that when applied to notions in physics that a change in n will produce a small incremental change in the derived results.

Of more immediate importance to physics, we discovered from the development of the mathematics, is that any derived construction must be inherently even numbered because odd numbering is immediately divergent. That directly implies that the act of particle creation must be by pairs. It was that discovery that allowed me to finally construct a satisfactory thought experiment regarding fundamental particles. I had pursued the idea itself with unsatisfactory results prior to this insight for many years.

What this odd numbered divergence might mean in terms of natural objects in space is much harder to illumine but it certainly serves as a warning that all apparently stable cyclic systems are potentially way more divergent than common sense would imply. For example it can be inferred that an incoming body could possibly trigger the removal of a comparable body in an apparently stable orbital system without overly disturbing the rest of the system.

On a personal note, I come to this problem through my personal training focused in the field of applied mathematics, rather than through the lens of present day thinking in physics. In fact, I am also trying to initially avoid in this paper most of the present work in physics and even the language of physics because it is premature for the purpose of this paper and is certain to confuse the reader. I would dearly love to associate the fp pair in the next section with some identified particle in today’s nomenclature, except that is premature. I am avoiding the language of physics, including even the idea of energy, because all these words carry baggage and hard to shake assumptions that we all share.

I included Rektorys’ Survey of Applied Mathematics in my references not for a specific item, but because the book is an attempt to inventory by statement without proofs, every theorem having value in applications. I had a close formal encounter with each and every theorem listed in the book’s thirteen hundred pages save a handful, sufficient to satisfy my appetite.

More critically, I was privileged to attend a one year course on General Relativity conducted by Hanno Rund and David Lovelock on their work that eliminated the unsatisfactory mathematical assumption of linearity from the General Theory. Much discussion took place regarding how particle physics might be synthesized in terms of the General Theory. It was apparent to me and perhaps others that such had to be a function of the metric itself. Yet no such metric presented itself or even came close.

Therefore, what I am conducting in the next several pages is a thought experiment that is informed by the idea of the existence of such a metric and the imposed mathematical necessity of pair wise creation in a particle based universe. The next several pages represent a rigorous attempt to describe the creation of a particle based universe as far as it is possible to proceed without simulation.

I emphasize that this is a thought experiment aimed at constructing a universe in the classical sense and specifically avoiding for now as much as possible a possibly premature attempt to associate results with known empirical results. In particular, the additional non acceptance of the equivalence of the inverse of infinity to zero causes me concern when addressing other such models that do not or can not make that distinction.

Importantly, the mathematics of the cyclic function is sufficiently robust that it is plausible to map the effect of a particle out to many cycles and to also construct assemblages of such particles and map their effect on each other to a distance of many cycles. This can be done to a high level of resolution and possibly thousands of cycles by taking advantage of the algebra I developed in the paper for this purpose.

Therefore, not only can we simply imagine the structures and their possibilities, we can also calculate their effect at a distance and to simulate experiments on them with the technology available today.

This means that it is possible to construct geometric objects whose actions and reactions can be identified as possible analogs to the particles of present day thinking in physics. This is also another good reason to avoid any modern terminology until calculation has caught up to our imagination.

You will find my understanding of time in the particle framework to be unusual and this will need spending some time on. I am informed for example that a finite object reflects a finite time function directly related to the number of fp pairs involved. I assume that relationship to be simply linear. I also do not expect the reader to be immediately comfortable with these ideas.

One further conjecture suggested by the quasi crystalline nature of the fp pair and the implied tetrahedral construction geometry is that polyhedral solids have physical meaning. One such object has 600 vertices and thus 1200 fps. If one then makes the conjecture that two such objects can adjoin and share one fp pair, we obtain a coincidental analog to proton neutron pairing. The remaining smaller polyhedral objects in complete form or missing a fp pair or more is also worth modeling as possible plausible derivatives of the breakup of the larger object. In any case, we have a framework for simulation and with the metric a method of establishing comparable mensuration over a range of polyhedral solids.

I believe other writers have occasionally proposed a pseudo crystalline structure to the particles of physics, but it is not where I started from at all. Rather I started with a careful thought experiment that ended up with a quasi crystalline structure with important time inferences.

The next section is a rigorous thought experiment whose aim is to imagine a particle universe constructed from a single unique fundamental particle created in pairs and then modeled by application of the generalized cyclic function order n were n is the number of particles involved between two and N the number of particles in the universe.

Once that is complete we discuss the issue of gravitation and the Dirac hypothesis and show the Euclidean metric as a first approximation. We also discuss the inferences of the generalized metric in light of the non zero nature of the inverse of infinity in our model.

Please Note: This model universe is meant to be understood as a thought experiment that may on the application of the implied calculation have physical meaning, but should not be applied as an analog to the standard theory until then. I also drop the n in my notation using cyclic functions when N is implied for the total particle count of the universe.

Defining the fundamental particle

Firstly, space S exists, and is conveniently described as a classic four-dimensional time-space manifold. Any object, whatever the word object may mean, contained in space S will induce curvature on space S. We leave the concept of existence to the purview of the philosophers.

Our postulated fundamental particle fp is bounded in three spatial dimensions by dd, which presents as a constant within the universe of fundamental particles U.

We see no reason at this point to assume that there is more than one such type of particle.

The particle fp must also signal its existence. We can imagine this happening as the turning on and off of a light switch. In this case it is reasonable to assume that the particle snaps in and out of existence over a time constant dt. This process continuously induces curvature on space in a cyclic manner transmitting the information of the fp’s existence.

A visual metaphor for a fundamental particle is a strobe light. By flashing on and off, it informs surrounding space of it location at light speed.

We now have two options. Either the fp is at rest and only moves relative to available curvature or alternatively, it recreates itself in the direction of curvature. In the first instance, the particle can establish variable velocities while transmitting curvature at c = dd/dt, a natural constant. In the second instance fp is itself traveling at c = dd/dt, but is changing direction every dt in response to the curvature signals emitted by adjacent particles in particular and that of the universe in general. I accept the second case, which puts the movement of all fp’s and their transmitted curvature at the speed of light and makes the first option as an unnecessary assumption and complication.

This second case also gives us an intuitive definition for inertia since motion for a finite set of fps cannot be changed smoothly as an external force F is continuously applied. Of course the incremental change element dd is so small that the observed effect is practically the same as with the calculus concept of mathematical continuity dependent of the concept of mathematical infinity. Infinity is something that the universe has no need to know and we can rigorously exclude it as an assumption.

In either case, the constancy of the speed of light is a direct result of the existence of the fp. This is comforting. Can we now construct our universe from this minimalist beginning?

The fps effect on curvature can be imagined as a push pull event, not dissimilar to a sine wave. The scaling unit dd will establish the apparent wavelength of this changing curvature, at least close to origin. A fp may be imagined as a bounded partition of space S. S can be imagined as an unimaginatively large sphere described by the expanding curvature surface generated by the creation of the first fp. This directly implies that every fp contained in space S is simultaneous since space S does not have a time clock independent of the time clock impressed by the fps contained in S. This also means explicitly that if and when a new fp is created, that all other fps will ‘know’. Time and distance, as we understand it, is a metric induced on space S by the universe of existing fps, but is not an external condition for S.

Since we are not inclined to let our fundamental particle fp slow down or sit still, we now need to construct a thought experiment for ‘at rest’ phenomena. Now, a single fp traveling in a straight line will generate a trailing cone of curvature. This is interesting but not useful. If instead we imagine two fps in close proximity, we can readily imagine a roughly synchronized dance in which the fps switches direction every dt cycle. The direction change is determined by the other fp’s curvature signal. The general shape of the configuration is tetrahedral and should converge to a fully synchronized dance in the absence of any external curvature change. In this thought experiment the fp pair traces a path around the tetrahedron, hitting each apex in turn while separated by one apex.

An fp pair is created by the twisting of space to form two bounded and adjacent fp’s. Each fp follows a step wise path in which it’s direction is determined principally by the curvature wave previously generated by it’s partner. The only stable configuration possible is the tetrahedron.

Without externally applied curvature, this pair is naturally at rest. Observe that the equivalent of the centre of mass will switch back and forth over a distance of dd/√2 thanks to the geometry of a tetrahedron.

A good visual analogy is to imagine a balloon receiving a half twist to form two spheres, then untwisting and disappearing, reappearing immediately and repeating the procedure at right angles to the first configuration. This cycle is then repeated over and over.

A more correct visualization is to imagine space twisting into existence on one edge of the tetrahedron creating two fps and then twisting back out of existence over the duration of dt. Thereupon the exact same process is reenacted on the opposing edge as a result of the curvature generated by the first event. This process repeats continually.

One immediate result of such a configuration is the generation of directional curvatures ‘dc’ in the four directions corresponding to the edges of the tetrahedron consisting of a repetition of 010101. This sets up and permits the possibility of additional pairs been drawn into the dance and linked together in a highly symmetric and crystalline-like formation. We can postulate the formation of stable geometric formations continuously emitting strong directional curvatures as well as establishing a general curvature on the surrounding space reflecting the fp content of the construct. These structures will be referred to as fp constructs for the purpose of this paper. An appropriate symbol could be P_2m , m been the number of fp pairs.

This 2-fp construct has all the implied symmetry of a tetrahedron as well as two opposite axis of rotation for the motion of the fps. This immediately implies that larger constructs will vary in regard to the axis of rotation chosen. Specifically, the tetrahedron has four axis of symmetry associated with each apex and the center of the opposite side, and two axis of symmetry through the center of opposing sides. These axis are not uniquely different inasmuch as an observer will be unable to separate the individuals in the two types of symmetry. However, the movement of fps from axis to axis occurs in either a clockwise or counter clockwise direction. These are separable to an observer. A 2-fp pair can thus present six unique aspects to an observer and to another pair.

These are: Apex on, clockwise path

Apex on, counter clockwise path

Face on, clockwise path

Face on, counter clockwise path

Edge on, clockwise path

Edge on counter clockwise path

This construct can obviously be further modeled and evaluated in the event that it is globally rotating as a result of externally applied curvatures.

An excellent intuitive conceptualization of the creation of an fp pair is to imagine taking a balloon and giving it a half twist thereby producing two spheres with a minimum effort. This may also be a useful conceptionalisation of the first act in the creation of the universe. An ideal symbol for such a pair is ironically . Direction change, rather than been thought of as from apex to apex, can now be better thought of as moving from edge to edge over the distance dd/√2.

A program to investigate the likely configurations requires extensive computer based simulation, which is now possible. Intuitively, we anticipate that it will be possible to construct objects that are analogs to our known particles. Inherent symmetry is built in and we see an affinity for directed curvatures that will provide an obvious analog for the forces holding the constructed objects together.

Conceptually we postulate the creation of directed curvatures that will have a stronger effect at short range than the balance of the generated curvature. Recall that during each cycle the fp generates a sphere of expanding curvature traveling at light speed. The directed curvature is a small partially bounded subset that has combined two fields in a straight line. We postulate that this line of curvature can link with the line of curvature of another pair and that this combination tends toward reinforcement and stability. This gives us an analog to a resonating waveform anchored on the two fp-pairs. What this will look like when we attempt to mathematically model it is unknown at this time.

One other intuitive concept that emerges from our thought experiments is that directed curvatures or waveforms or photons have an affinity to combine and resonate where conditions permit. The rules of such events are only hinted at. More positively, our fp pair mode establishes ground rules for the generation of these formations and will allow us to create working models that permit the understanding of the nature of their linkages and boundedness.

The linkage of two 2-fp constructs is possible on any of the four edgewise axis and perhaps the other axis presented and generating a signal sig equal to 010101… In line we have a combined sig ladder between the two 2-fps that tends to keep them in place. If they move slightly out of position then the next passage of the sig will jostle them back into position. Oscillation is likely the norm. The distance between the two 2-fp constructs can vary greatly with binding strength obviously increasing as proximity improves.

Extending this logically, we can postulate the formation of a larger tetrahedral construct consisting of a central 2-fp with four 2-fps linked at the apexes. Varying linkage distances and synchronicity generates a variable 10-fp construct with multiple acceptable unique configurations to be tested by computer simulation.

Taking this thought experiment further we must consider the existence of assemblages of sigs been emitted and absorbed by the fp constructs. Such assemblages must react with the contained fps in the construct for both emission and absorption. These will perhaps be the analogs to photons and binding forces.

Photons are semi-bounded assemblages of curvatures moving in one direction. It is possible that a sufficiently energetic photon is the precursor to a fundamental particle fp. Photons do not impress new curvature on space S in the same sense as an fp outside that curvature already contained by the photon but they may react with each other.

What becomes clear from the forgoing discussion is that we can readily model n-fp constructs of large n. Stability will vary depending on the level of internal oscillation and shell completeness. Whether we can now build out our current knowledge of particle physics using these simple foundations can only be answered successfully through extensive computer modeling.

Time inference

These n-fp constructs will impress their own general curvature and inherent time constant on their surrounding local space L. L – Space is defined as a domain containing a countable subset of fp pairs.

We have postulated by a simple axiom of existence that each fp reflects a dt which is a universal constant perceived as dependent only on the number of fps in the universe. It is also clear that any n-fp construct or more generally any subset of fps will generate its own internal time scale tau. This is an inevitable inference of our definition for S in which the universe of all fps impresses time and space dimension on the universe. This means that any subset of fps contained within the universe has the same set of rules applying as is true for the whole.

We are stating that both the universe and any defined subset will exhibit a time value tau that is dependent on the fp content. This is not to say that a subset will not also be part of the universal tau, it obviously must, but that for a given subset, the effect of the universal tau is small enough to be ignored. I anticipate that for particle physics, the tau may be an analog of spin. I have no particular suggestions on how to construct a derivative equation before enough modeling is done to put us firmly on track.

I am saying, however, that constructs of varying fp content will exhibit an internal clock that will vary as to the fp count.

This puts the classical understanding of time on its head. Our philosophy has informed us that time is uniform and somewhat independent. We are saying instead that our sense of time is a direct result of the impress of the existence of a universe of simultaneous fundamental particles that are bounded regions of space described by dt and dd. In addition we are saying that in any subset of fps, it is convenient to treat that subset as a small separate universe with an internally consistent resultant time component tau.

Large-scale constructs

Without the benefit of direct modeling, we are way out on a limb at this point. We can intuitively speculate from the forgoing that large assembled n-fp constructs could have the ability to absorb and re emit photons while effectively expanding or contracting. We can intuit the existence of a strong induced curvature around the multi proton construct of an atom that will behave similarly to a series of shells that permits the holding of an electron analog anchoring its photon ring. We intuit that our push-pull induced curvature is quite capable of creating the known electromagnetic forces. Without large scale modeling, we cannot know precisely how.

We can deduce that any form of asymmetric curvature both internal and externally induced on the construct will result in physical spin. This will have a large effect on the form of the induced curvature and the form of the construct’s interaction with other constructs.

We have described the likely characteristics of a fundamental particle and its derivative constructs. The design principle is clearly rich enough to permit success in modeling the full universe of particles and forces known to physics. And we have done this without introducing a single new law of physics outside of insisting that a fundamental particle exists and executing a thought experiment about its nature.

I suspect that these particular characteristics can be proven rigorously as both necessary and sufficient for the existence of the universe. The one remaining task left to us is to assign a mathematical function to describe the fps in the universe in order for us to generate useful information and predictions.

Mathematics throws up a wide range of potentially useful cyclic functions. Beside the old standbys of sine and cosine, we have Taylor series and any number of less convenient non-functions and similar constructs. None of these functions reflect the fp content of the universe and will be at best a stopgap for the sake of modeling the theory and could possibly lead to serious error and/or simple distortion.

Fortunately, we have the perfect function(s) at hand in the form of the generalized cyclic functions we have just introduced. These functions behave conformably for all n > 2. The shape of the curve and apparent wavelength are extremely similar for any large n>>>2 and certainly tend to converge to extreme similarity for both large n and large x. This permits stability in the overall structure of the universe and its laws. A shift in the fp content of the universe will have a minute effect on the universe and its apparent laws under these equations.

Mathematics of the fp

We have described the fp as a strobe light generating shells of alternating positive and negative curvature on space. We have also pointed out that each fp reflects the total number of fps in the universe and that this is true simultaneously.

In the case of a universe of fp pairs, it is consistent to model their behavior with the sine function. In the case of four fps, it is consistent to model their behavior with C(4,0) which we discussed earlier. We can apply this approach in our modeling for any n-fp construct

More generally we can model all fp particles in the universe with the cyclic function C(n,0). For our use, we know that there exists a large number of fps in our universe that is believed to number around 10 to the power of 78. It is therefore convenient to think of the fp as been described by the function C(10⁷⁸,0). Symbolically, it is convenient to use the following form of C(∞,0), although we risk confusion with the concept of mathematical infinity. The universe does not know what mathematical infinity is. This does suggest a natural way to convert mathematical calculus into physical calculus and that it should be done at least as a useful exercise. We may surprise ourselves and gain greater insight into the concept of infinity.

In practice, we will be able to model fp constructs that will range up to perhaps several hundred fp pairs using the sine function alone losing only accuracy.

From the mathematics of the generalized cyclic function we can make one other key inference. The odd ordered functions are uniformly divergent in the negative suggesting sharp changes in behavior. This implies that it is a physically necessary for new fps to form two at a time as we have described.

It appears reasonable to postulate that these pairs can be formed as a result of the collapse or folding of a semi bounded photon construct that carries sufficient curvature. We can also postulate that any two photons containing enough curvature can interact and form a 2-fp pair while carrying of surplus curvature in the form of another photon. We might speculate that this can happen in a low curvature environment where linkage is not interfered with over some vast distance as well as in a high curvature environment, neither of which applies on earth. One other such environment is implied as a condition for the postulated fractal like surface of the universe.

I observe that if we assume that the universe had a beginning, then a direct logical outcome is that the universe can be described as a sphere expanding at the speed of light from the observer’s perspective. I understand no other logical derivative of the assumption of existence at this time.

The explicit necessity of pair creation also tells us that we can focus much of our work on the behavior of the fp pair with the assistance of our old friends the sine and cosine functions. This is important, since short range function convergence will become progressively more difficult with computer simulation of larger and larger n-constructs. If instead our protocol for investigation is designed around using sine functions to simulate a large n structure made up of many pairs, then it is merely a matter of substituting the nth order cyclic function in step wise fashion in order to observe the tightening up of the n structure (it will get much smaller) and to arrive at a true model (or close approximation).

Implied cosmology

We have directly linked the particle content of the universe to the equation describing any fp. We have a number of inferences from this approach.

1. Cosmological content is a direct result of the generalized cyclic equation describing the fundamental particle. The space-time continuum or manifold impressed by this content is as understood in general relativity, which is minimalist in design. The number of fps in the universe, the size of the fp, the apparent age of the universe and the speed of light itself are all directly linked through this formulation.

2. Dirac’s large number hypothesis is a natural result of this formulation. Our impressed Tau for the universe of all particles will be the apparent age of the universe, whatever that may mean. The association of dt and dd and the total number of fps in the universe within the founding equation impresses the observed scaling system that generates the large number hypothesis and is excellent evidence that we may be on to something. This means explicitly that the apparent size of the observed particle system is a function of the number of such particles within the universe and this number dictates the apparent time scale of the universe.

3. It is reasonable that gravity G can be precisely defined. The curvature H(r) derived from a single particle can be described by the following equation:

The equation (43)

holds for any fp and by simple extension for any fp construct. For simplicity, we are assuming the Newtonian space metric where the geometric component is the inverse square law. Calculating H in local space for a fp pair presents a special difficulty because the switching of position creates two points of origin for any net calculation at a remove from the pair. The distance term r is measured as the number of dd’s counted from the particle to the observer.

Gravitational theory

We have noted that the applied curvature effect of a fp can be described by the equation

(44)

where n is the number of fps in the universe and we assume a simple inverse square law over distance.

We have also noted that locally, that matter is dominated by fp-pairs. This means that the second order cyclic form will dominate and we can simplify our lives greatly in terms of calculating the larger scale effects by working specifically with the second order form. This cannot be accomplished at the particle assemblage level in which the second order model will only provide a first approximation.

The gravitational force of a large object of mass M can be approximated by the simple expedient of calculating the net effect of the contained fps using H(r). At a significant distance this will vary linearly with H(r) as the implied geometric effects of M diminish. Again we can reasonably expect that the effects of 2-fp geometry will be dominant, as this is again local space. Formally, M varies as to the volume of M, which can be described in the case of a sphere as 4/3πR³ in which R is the radius. Obviously M could be simply replaced by M defined as the number of fps and R could be replaced by R defined as the number of fp wavelengths required to measure the radius of the sphere.

We will first start our investigation by demonstrating the resultant calculation of gravitational force in Newtonian Physics. Therein gravity varies as to 1/r² by the inverse square law. Thus for any particular interval {a, b}, the total applied effect varies as to the over that interval. Since the indefinite integral , we determine over the interval {a, b} the resultant value of ( -1/b + 1/a ) or (b-a)/ab.

If we now set and , we have the resulting value of . In Newtonian Physics gravity will vary as to over the interval 

We now use our equation of the second order cyclic function which coincides with the dominant fp pair structure of local space. Since this function has a cyclic component we will calculate our integral over the interval . You will observe that 1/r² is still a dominant part of this equation.

The indefinite integral of C(2,0)r/r² or cos(r)/ r² is as follows: (45)

)

This is convergent. By the by, we do not know what it is properly convergent too. This is one of those nasty problems in mathematics. It goes without saying that the higher nth order versions will be just as aggravating and uncooperative.

Calculating this integral between the interval , we have the following results for the first term:

(46)

This component is exactly the same as the Newtonian solution just shown. This happens to be excellent news since the most influential component locally is immediately producing the Newtonian metric.

What does the rest of the equation contribute? Here we have the aforementioned problem. It has been proven that this component is convergent. It has never been shown what it is convergent to. It is a very small number approaching zero but not necessarily zero. Certainly, it is close enough to zero in local space to not matter and to be effectively undetectable above the noise.

And as r increases it is reasonable that if the convergent value of the second non-Newtonian component is non-zero then the resultant effect of this component will increase. More explicitly, the first Newtonian term is converging on zero much faster that the non Newtonian second term. This means that at some point for a sufficiently large enough r the effect of the second component will dominate the effect of the inverse square component.

Since we have already eliminated mathematical infinity as an assumption we can state that both the Newtonian component and the non-Newtonian component will independently converge to a small positive number identical with value for physical infinity ∞ = 10^-78.

This allows us to postulate that the net effect of gravity is possibly twice as strong as predicted by classical Newtonian physics in the inter Stellar and inter Galactic void. This also means that a significant part of the missing mass problem is a direct artifact of the erroneous application of mathematical infinity. Applying this new formulation may account for all the missing mass even before we factor in any possible effect of the higher ordered cyclic function formulation that more appropriately reflects the particle count in the universe.

Returning to the indefinite integral is:

(47)

This equation will be important in mapping curvature in proximity to objects such as particles and atoms with a countable number of fp pairs. It will not however give us the same easy ride that the fp pair gave us by simply resolving into a Newtonian component and an extra component. Here we still have two components with the same cosmological result and the fact that matter is made up of 2 fp pairs imposes the dominant Newtonian locally.

Again for the first term over the interval, were n is unrelated to the order of the cyclic function we have the following after setting C₀r = C(n, 0)r

(47)

Which is hardly presenting the simple resolution seen for n = 2. The second term gives us:

(48)

Finally we can readily combine the two terms giving us an equation describing the curvature in the vicinity of a particle with many fp pairs.

(49)

Net curvature for the fp pair

The fp pair traces out a tetrahedron as it switches back and forth from edge to edge. The center of influence is the half way point of this switching path and can be used as a point of origin for calculation purposes. Any two apexes to point of origin lines can be used as axis for setting up the curvature equations for any point in space surrounding the fp pair. A more useful axis is the unique line between the mid points of the edges formed by the successive cycles of the fp pair. The other two edge pairs are excluded because they do not represent the appearance of both fps on the same edge. This can be naturally called the polar axis.

This formulation is not overly convenient for mapping the important edgewise behavior, so a second formulation is likely justified using an apex as a point of origin. The more important difficulty is accommodating the switching back and forth of the fp pair which is better facilitated by the use of the polar axis.

In any event we can construct a vector equation for the net curvature at point P(r, λ, δ) consisting of the individual contributions from the four apexes using s = √3/8 as the distance of any apex from the origin and λ and δ as the angles based on the polar axis and a line to an apex rotating on the polar axis.

4 Cosmological Red Shift. Another immediate artifact of using the higher order cyclic function formulation is the fact that the apparent wavelength w(x) is declining over great distances since w(x) is converging to a constant as x becomes very large. This means that distant objects are really closer than would be implied by the natural assumption of uniformity. This implies that the observed Galactic red shift is primarily a result of this effect rather than the continuous creation of new fps. Of course both effects could be contributing, except that new fp production is declining inversely to the total volume of the universe. Our model does allow new fp pair production, which would lead to an apparent red shift due to the general increase in particle content. A simple interpretation implies a set age for the universe linked to the number of fps. The real possibility of production and consumption of fp pairs been possibly in general balance throughout the universe will interfere with any nice speculations we may wish to make regarding the age of the universe.

More precisely, for any photon we know that E = hc/λ. For any photon assemblage we also now realize from our knowledge of cyclic functions that the impressed dd component of the photon is shrinking because w(x) is converging to a constant as the photon gets further from its point of origin.

Since E = hc/ λ = hdd/dt λ and since E and dt (the observed time structure of the photon will not change) are held immutable in our measuring regime, this effect can only be observed as a lengthening of the wavelength λ. This is the most likely origin of the observed universal red shift.

5 Schwarzschild Event Horizon. The much discussed interpretation of the Schwarzschild solution is a natural outcome of mathematical infinity and is meaningless except to predict tight packing of particle content. A more obvious interpretation based on our understanding of fp pairs is that two pairs in close proximity may be able to be able to create a new fp pair while releasing the other pair in the form of partially bounded curvatures. This carries off half the gravity associated with the two pair.

While this is taking place, the partially bounded curvatures may be expected to recombine into various photonic forms and migrate out of the star carrying off the associated curvature content. We can expect a resultant spectrum in which a huge amount of energy is been discharged right across the spectrum. In this form, the bulk of the energy associated with a star will be carried outside the gravity well created by the containing Galaxy. A portion of this energy may also reform into fp pairs in the low curvature environment of space. Most of this can expected to also be outside the gravity well of the galaxy.

The terminal content of gravity wells can thus be an analog to a packed 3D fractal like set that is subject to analysis as such.

Conjecture: Quasars are event horizon phenomena and are likely way closer than presently assumed. Some may even be in our Galaxy. Massive jets of photonic energy or partial curvature carries off content and gravity and these jets are seen as some of it decays back into visible matter.

Summary

We have constructed a concept of the universe using only the assumption of existence and the insight derived from the mathematics of the generalized cyclic function, without reference yet to the extensive body of empirical information. A next step is to apply simulation methods to the known range of regular polyhedral objects in which each vertice is occupied by a fp pair tracing a tetrahedral path and calculate their mutual effects on each other using the new metric. With the results of that work it will be possible to produce comparable calculations over the full range of possible objects and determine if there is any apparent similarity to the empirical data. We should face no particular difficulty in simulating polyhedral objects with even thousands of vertices using the algebra and achieving extreme precision in calculation.